3x-4y+2z+8=0, x+5y-3z+2=0, 5x+3y-z+6=0

Get the constants on the right:

3x-4y+2z=-8, x+5y-3z=-2, 5x+3y-z=-6

I'll color the constants red:



 

x is the FIRST unknown and so  is just like 
except that the FIRST column is replaced by the three red numbers: 



y is the SECOND unknown and so  is just like 
except that the SECOND column is replaced by the three red numbers: 




z is the THIRD unknown and so  is just like 
except that the THIRD column is replaced by the three red numbers,



Do you know how to evaluate a 3x3 determinant?
If not, post again asking how to.  I will assume
you already know how.






Then the solutions are:





Edwin